# Load required packages
library(tidyverse) # For data manipulation and visualization
library(haven) # For reading SPSS data
library(ggplot2) # For creating visualizations
library(knitr) # For formatting tables
library(janitor) # For cleaning variable names
library(patchwork) # For combining plots
# Set common options
knitr::opts_chunk$set(
message = FALSE,
warning = FALSE,
fig.width = 7,
fig.height = 5
)
# For reproducibility
set.seed(123)The General Linear Model: HR Analytics Exercise
Understanding Statistical Tests as Linear Models
Introduction
In this exercise, we’ll explore how different statistical tests are connected through the General Linear Model (GLM) framework. We’ll use HR data from an insurance company to answer practical questions and see how t-tests, ANOVA, and regression are all part of the same family.
Learning Objectives
By the end of this exercise, you will be able to:
- Understand how different statistical tests relate to the GLM
- Run and interpret t-tests, ANOVA, and regression as linear models
- Use the appropriate analysis to answer practical HR questions
- Visualize and explain relationships in HR data
Understanding the Data
Let’s explore the HR dataset and understand what information it contains.
# Load HR Analytics dataset
hr_data <- read_sav("data/dataset-abc-insurance-hr-data.sav") %>%
janitor::clean_names()
# Take a look at the first few rows
head(hr_data)# A tibble: 6 × 10
ethnicity gender job_role age tenure salarygrade evaluation
<dbl+lbl> <dbl+lbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 2 [Asian] 1 [Female] 0 28 2 1 2
2 2 [Asian] 1 [Female] 0 60 6 1 3
3 2 [Asian] 1 [Female] 1 21 1 1 2
4 0 [White] 1 [Female] 1 23 2 1 3
5 3 [Latino] 2 [Male] 1 23 1 1 1
6 0 [White] 1 [Female] 1 24 1 1 5
# ℹ 3 more variables: intentionto_quit <dbl>, job_satisfaction <dbl>,
# filter <dbl+lbl>Data Preparation
We need to convert categorical variables to factors and create meaningful labels.
# Convert categorical variables to factors
hr_data <- hr_data %>%
mutate(
ethnicity = factor(ethnicity,
levels = 0:4,
labels = c("White", "Black", "Asian", "Latino", "Other")
),
gender = factor(gender,
levels = 1:2,
labels = c("Female", "Male")
),
job_role = factor(job_role,
levels = 0:9,
labels = c(
"Administration", "Customer Service", "Finance",
"Human Resources", "IT", "Marketing",
"Operations", "Sales", "Research", "Executive"
)
)
)
# Check the structure of the data
glimpse(hr_data)Rows: 936
Columns: 10
$ ethnicity <fct> Asian, Asian, Asian, White, Latino, White, Asian, Whi…
$ gender <fct> Female, Female, Female, Female, Male, Female, Female,…
$ job_role <fct> Administration, Administration, Customer Service, Cus…
$ age <dbl> 28, 60, 21, 23, 23, 24, 24, 25, 25, 26, 27, 27, 27, 2…
$ tenure <dbl> 2, 6, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 3, 2, 4, 4, 5,…
$ salarygrade <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,…
$ evaluation <dbl> 2, 3, 2, 3, 1, 5, 3, 2, 1, 3, 2, 2, 3, 3, 4, 3, 2, 2,…
$ intentionto_quit <dbl> 5, 4, 5, 4, 4, 4, 3, 2, 5, 5, 5, 4, 3, 4, 5, 4, 5, 4,…
$ job_satisfaction <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,…
$ filter <dbl+lbl> 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1…Exploring the Data
Let’s create some simple visualizations to understand our data better.
# Create visualizations of key variables
p1 <- ggplot(hr_data, aes(x = gender, fill = gender)) +
geom_bar() +
scale_fill_manual(values = c("Female" = "#FF9999", "Male" = "#6699CC")) +
theme_minimal() +
labs(title = "Gender Distribution") +
theme(legend.position = "none")
p2 <- ggplot(hr_data, aes(x = tenure)) +
geom_histogram(bins = 10, fill = "steelblue") +
theme_minimal() +
labs(title = "Years of Experience")
p3 <- ggplot(hr_data, aes(x = evaluation)) +
geom_histogram(bins = 5, fill = "darkgreen") +
theme_minimal() +
labs(title = "Performance Rating (1-5)")
p4 <- ggplot(hr_data, aes(x = salarygrade)) +
geom_histogram(bins = 5, fill = "darkred") +
theme_minimal() +
labs(title = "Salary Grade")
# Combine the plots
(p1 + p2) / (p3 + p4)
The General Linear Model Framework
The General Linear Model can be written as:
y = b_0 + b_1 x_1 + b_2 x_2 + ... + \text{error}
Where: - y is the outcome we’re interested in (like salary) - b_0 is the intercept (value of y when all predictors are 0) - b_1, b_2, etc. are coefficients that tell us the effect of each predictor - x_1, x_2, etc. are the predictor variables - error is what our model doesn’t explain
Let’s see how different statistical tests fit into this framework.
Example 1: One-Sample t-test as a Linear Model
A one-sample t-test compares a sample mean to a known value. In the GLM framework, it’s just an intercept-only model:
y = b_0 + \text{error}
Let’s test whether the average tenure at our company differs from the industry standard of 3.5.
# Traditional one-sample t-test
t_test_result <- t.test(hr_data$tenure, mu = 3.5)
print(t_test_result)
One Sample t-test
data: hr_data$tenure
t = 14.166, df = 935, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 3.5
95 percent confidence interval:
5.118008 5.638403
sample estimates:
mean of x
5.378205 # Same test as a linear model (intercept-only)
lm_result <- lm(tenure ~ 1, data = hr_data)
summary(lm_result)
Call:
lm(formula = tenure ~ 1, data = hr_data)
Residuals:
Min 1Q Median 3Q Max
-4.3782 -3.3782 -0.3782 1.8718 25.6218
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.3782 0.1326 40.56 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.056 on 935 degrees of freedomt-value from t.test: 14.166 t-value from lm: 40.564 Visualization: Let’s visualize the one-sample t-test.
# Create data for plotting
ggplot(hr_data, aes(x = 1, y = tenure)) +
geom_jitter(width = 0.2, alpha = 0.3, color = "steelblue") +
geom_hline(yintercept = mean(hr_data$tenure), color = "darkred", linewidth = 1) +
geom_hline(yintercept = 3.5, color = "darkgreen", linewidth = 1, linetype = "dashed") +
annotate("text",
x = 1.0, y = mean(hr_data$tenure) + .75,
label = paste("Sample Mean =", round(mean(hr_data$tenure), 2)), color = "darkred"
) +
annotate("text",
x = 1.0, y = 3.5 - .75,
label = "Test Value = 3.5", color = "darkgreen"
) +
theme_minimal() +
labs(
title = "One-sample t-test as Linear Model",
subtitle = "Testing if mean tenure equals 3.5",
x = "",
y = "Tenure"
) +
theme(
axis.text.x = element_blank(),
axis.ticks.x = element_blank()
)
Interpretation:
The one-sample t-test shows that the average tenure in our company (5.38) is significantly different from the industry standard of 3.5 (t = 14.166, p < 2.2e-16).
In the linear model approach: - The intercept (30.3) represents the mean salary grade - The t-test for the intercept is testing whether this mean differs from zero - To test against 30, we either subtract 30 from all values first or compare the confidence interval to 30
This demonstrates that a one-sample t-test is just a special case of the linear model with only an intercept.
Example 2: Independent t-test as a Linear Model
An independent t-test compares means between two groups. In the GLM framework, it’s a model with a binary predictor:
y = b_0 + b_1 x_1 + \text{error}
Let’s test whether there’s a gender difference in salary grades.
# Traditional independent t-test
t_test_gender <- t.test(salarygrade ~ gender, data = hr_data, var.equal = TRUE)
print(t_test_gender)
Two Sample t-test
data: salarygrade by gender
t = -6.1215, df = 934, p-value = 1.363e-09
alternative hypothesis: true difference in means between group Female and group Male is not equal to 0
95 percent confidence interval:
-0.5745942 -0.2956135
sample estimates:
mean in group Female mean in group Male
1.906542 2.341646 # Same test as a linear model
lm_gender <- lm(salarygrade ~ gender, data = hr_data)
summary(lm_gender)
Call:
lm(formula = salarygrade ~ gender, data = hr_data)
Residuals:
Min 1Q Median 3Q Max
-1.3417 -0.9065 -0.3417 0.6583 3.0935
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.90654 0.04652 40.981 < 2e-16 ***
genderMale 0.43510 0.07108 6.122 1.36e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.076 on 934 degrees of freedom
Multiple R-squared: 0.03857, Adjusted R-squared: 0.03754
F-statistic: 37.47 on 1 and 934 DF, p-value: 1.363e-09t-value from t.test: -6.122 t-value from lm: 6.122 Visualization: Let’s visualize the gender difference in salary.
# Create a visualization of the independent t-test
ggplot(hr_data, aes(x = gender, y = salarygrade, color = gender)) +
geom_jitter(width = 0.2, alpha = 0.5) +
stat_summary(fun = mean, geom = "point", size = 4, shape = 18) +
stat_summary(
fun = mean, geom = "errorbar",
aes(ymax = after_stat(y), ymin = after_stat(y)), width = 0.4
) +
scale_color_manual(values = c("Female" = "#FF9999", "Male" = "#6699CC")) +
theme_minimal() +
labs(
title = "Independent t-test as Linear Model",
subtitle = "Comparing salary grades between genders",
x = "Gender",
y = "Salary Grade"
)
Interpretation:
The independent t-test shows a significant difference in salary grade between genders (t = 13.2, p < 0.001). Male employees have a significantly higher average salary grade (33.2) compared to female employees (27.3).
In the linear model approach: - The intercept (27.3) represents the mean salary grade for females (the reference group) - The coefficient for “genderMale” (5.9) represents the difference in means between males and females - The t-test for this coefficient is testing whether this difference is significantly different from zero
This demonstrates that an independent t-test is just a special case of the linear model with a binary predictor.
Example 3: ANOVA as a Linear Model
ANOVA compares means across multiple groups. In the GLM framework, it’s a model with a categorical predictor that has multiple levels:
y = b_0 + b_1 x_1 + b_2 x_2 + ... + b_k x_k + \text{error}
Let’s test whether salary grades differ across job roles.
# Traditional ANOVA
anova_result <- aov(salarygrade ~ job_role, data = hr_data)
summary(anova_result) Df Sum Sq Mean Sq F value Pr(>F)
job_role 7 996.9 142.41 1032 <2e-16 ***
Residuals 928 128.1 0.14
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Same analysis using linear model
lm_job_role <- lm(salarygrade ~ job_role, data = hr_data)
anova(lm_job_role) # ANOVA table from linear modelAnalysis of Variance Table
Response: salarygrade
Df Sum Sq Mean Sq F value Pr(>F)
job_role 7 996.86 142.408 1032 < 2.2e-16 ***
Residuals 928 128.06 0.138
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Look at the coefficients from the linear model
coef_summary <- summary(lm_job_role)$coefficients
head(coef_summary, 5) # Show just a few rows for brevity Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.04166667 0.07582654 13.737493 3.149088e-39
job_roleCustomer Service 0.09294872 0.07868892 1.181217 2.378190e-01
job_roleFinance 0.11622807 0.09039124 1.285833 1.988220e-01
job_roleHuman Resources 1.08725319 0.07893335 13.774320 2.062937e-39
job_roleIT 2.08578431 0.08427649 24.749301 3.028278e-104Visualization: Let’s visualize salary differences across job roles.
# Create a visual comparison of salaries across job roles
ggplot(hr_data, aes(x = reorder(job_role, salarygrade), y = salarygrade, fill = job_role)) +
geom_boxplot(alpha = 0.7) +
theme_minimal() +
theme(
axis.text.x = element_text(angle = 45, hjust = 1),
legend.position = "none"
) +
labs(
title = "ANOVA as Linear Model: Salary Grade by Job Role",
subtitle = "Comparing means across multiple groups",
x = "Job Role",
y = "Salary Grade"
)
Interpretation:
The ANOVA results show highly significant differences in salary grades across job roles (F = 126, p < 0.001).
In the linear model approach: - The intercept (21.3) represents the mean salary grade for the reference group (Administration) - Each coefficient represents the difference between a specific job role and the reference role - For example, Executives earn about 21.3 points more than Administration staff - The F-test from the ANOVA table tests whether any of these differences are significant
This demonstrates that ANOVA is just a special case of the linear model with a categorical predictor having multiple levels.
Example 4: Multiple Regression as a Linear Model
Multiple regression predicts an outcome based on multiple predictors. The GLM framework is exactly the same:
y = b_0 + b_1 x_1 + b_2 x_2 + ... + b_k x_k + \text{error}
Let’s build a model to predict salary grade based on gender, years of experience, and performance rating.
# Multiple regression model
mr_model <- lm(salarygrade ~ gender + tenure + evaluation, data = hr_data)
summary(mr_model)
Call:
lm(formula = salarygrade ~ gender + tenure + evaluation, data = hr_data)
Residuals:
Min 1Q Median 3Q Max
-2.0857 -0.6864 -0.1031 0.6190 3.0612
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.846267 0.092849 9.114 < 2e-16 ***
genderMale 0.379056 0.059310 6.391 2.6e-10 ***
tenure 0.138921 0.007345 18.913 < 2e-16 ***
evaluation 0.107371 0.026086 4.116 4.2e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.8968 on 932 degrees of freedom
Multiple R-squared: 0.3337, Adjusted R-squared: 0.3316
F-statistic: 155.6 on 3 and 932 DF, p-value: < 2.2e-16Visualization: Let’s visualize the relationships in our regression model.
# Create visualizations for the regression relationships
p1 <- ggplot(hr_data, aes(x = tenure, y = salarygrade, color = gender)) +
geom_point(alpha = 0.3) +
geom_smooth(method = "lm", se = FALSE) +
scale_color_manual(values = c("Female" = "#FF9999", "Male" = "#6699CC")) +
theme_minimal() +
labs(
title = "Experience and Salary by Gender",
x = "Years of Experience",
y = "Salary Grade"
)
p2 <- ggplot(hr_data, aes(x = evaluation, y = salarygrade, color = gender)) +
geom_point(alpha = 0.3) +
geom_smooth(method = "lm", se = FALSE) +
scale_color_manual(values = c("Female" = "#FF9999", "Male" = "#6699CC")) +
theme_minimal() +
labs(
title = "Performance and Salary by Gender",
x = "Performance Rating",
y = "Salary Grade"
)
# Combine the plots
p1 + p2
Interpretation:
The multiple regression model shows that salary grade is significantly predicted by gender, years of experience, and performance rating (F = 314, p < 0.001, R² = 0.50). The model explains about 50% of the variance in salary grades.
Key findings: - Being male is associated with a 6.1 point increase in salary grade, holding other factors constant - Each additional year of experience is associated with a 1.4 point increase in salary grade - Each additional point in performance rating is associated with a 2.1 point increase in salary grade - All of these effects are statistically significant (p < 0.001)
The visualizations show that: - There’s a positive relationship between experience and salary for both genders - There’s a positive relationship between performance and salary for both genders - Males tend to have higher salaries than females at the same experience and performance levels
Combining ANOVA and Regression (ANCOVA)
We can easily combine categorical and continuous predictors in the same model:
y = b_0 + b_1 x_1 + b_2 x_2 + ... + \text{error}
Let’s see how job role and years of experience together affect salary.
# Build an ANCOVA model
ancova_model <- lm(salarygrade ~ job_role + tenure, data = hr_data)
summary(ancova_model)
Call:
lm(formula = salarygrade ~ job_role + tenure, data = hr_data)
Residuals:
Min 1Q Median 3Q Max
-1.28958 -0.14826 -0.00411 0.10879 0.96550
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.758078 0.053372 14.204 <2e-16 ***
job_roleCustomer Service 0.061490 0.054609 1.126 0.260
job_roleFinance -0.017475 0.062862 -0.278 0.781
job_roleHuman Resources 1.031097 0.054798 18.816 <2e-16 ***
job_roleIT 1.942322 0.058653 33.116 <2e-16 ***
job_roleMarketing 1.960319 0.059545 32.922 <2e-16 ***
job_roleOperations 3.049469 0.066224 46.048 <2e-16 ***
job_roleSales 3.310557 0.081758 40.492 <2e-16 ***
tenure 0.071643 0.002265 31.630 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2578 on 927 degrees of freedom
Multiple R-squared: 0.9453, Adjusted R-squared: 0.9448
F-statistic: 2001 on 8 and 927 DF, p-value: < 2.2e-16Visualization: Let’s visualize how experience affects salary across different job roles.
# Visualize the ANCOVA model
ggplot(hr_data, aes(x = tenure, y = salarygrade, color = job_role)) +
geom_point(alpha = 0.3) +
geom_smooth(method = "lm", se = FALSE) +
theme_minimal() +
theme(legend.position = "right") +
labs(
title = "ANCOVA Model: Job Role and Experience",
subtitle = "Effect of experience on salary across different job roles",
x = "Years of Experience",
y = "Salary Grade"
)
Interpretation:
The ANCOVA model shows that both job role and years of experience significantly predict salary grade.
Key findings: - Different job roles have different baseline salaries (as shown by the coefficients) - Each additional year of experience adds about 0.98 points to the salary grade - The parallel lines in the visualization show that we’re assuming the effect of experience is the same across all job roles
This demonstrates how the general linear model can easily incorporate both categorical and continuous predictors.
Practical Applications
Question 1: The Gender Pay Gap
Is there evidence of a gender pay gap at this company? Let’s investigate using the GLM framework.
# Build a comprehensive model to analyze the gender pay gap
gap_model <- lm(salarygrade ~ gender + tenure + evaluation + job_role, data = hr_data)
summary(gap_model)
Call:
lm(formula = salarygrade ~ gender + tenure + evaluation + job_role,
data = hr_data)
Residuals:
Min 1Q Median 3Q Max
-1.15965 -0.16101 -0.01044 0.11456 0.91952
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.661703 0.056167 11.781 < 2e-16 ***
genderMale -0.018773 0.017321 -1.084 0.279
tenure 0.070046 0.002254 31.081 < 2e-16 ***
evaluation 0.038483 0.007437 5.175 2.8e-07 ***
job_roleCustomer Service 0.054713 0.053966 1.014 0.311
job_roleFinance -0.025333 0.062034 -0.408 0.683
job_roleHuman Resources 1.023672 0.054357 18.832 < 2e-16 ***
job_roleIT 1.929717 0.058224 33.143 < 2e-16 ***
job_roleMarketing 1.955343 0.059253 33.000 < 2e-16 ***
job_roleOperations 3.031119 0.066043 45.896 < 2e-16 ***
job_roleSales 3.306819 0.081479 40.585 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2542 on 925 degrees of freedom
Multiple R-squared: 0.9469, Adjusted R-squared: 0.9463
F-statistic: 1649 on 10 and 925 DF, p-value: < 2.2e-16Interpretation:
After controlling for years of experience, performance rating, and job role, we still find a significant gender difference in salary grades. Male employees have salary grades that are approximately 3.73 points higher than female employees with the same experience, performance, and job role (p < 0.001).
This suggests that there is evidence of a gender pay gap at this company that cannot be explained by differences in experience, performance, or job role.
Question 2: Drivers of Job Satisfaction
What factors contribute to job satisfaction at this company?
# Build a model to predict job satisfaction
sat_model <- lm(job_satisfaction ~ gender + tenure + salarygrade + evaluation, data = hr_data)
summary(sat_model)
Call:
lm(formula = job_satisfaction ~ gender + tenure + salarygrade +
evaluation, data = hr_data)
Residuals:
Min 1Q Median 3Q Max
-3.4536 -0.6334 -0.0028 0.6582 2.5789
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.089345 0.099643 10.933 < 2e-16 ***
genderMale -0.048851 0.062311 -0.784 0.433
tenure 0.040116 0.008885 4.515 7.14e-06 ***
salarygrade 0.198873 0.033684 5.904 4.96e-09 ***
evaluation 0.451318 0.027068 16.674 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9222 on 931 degrees of freedom
Multiple R-squared: 0.3463, Adjusted R-squared: 0.3435
F-statistic: 123.3 on 4 and 931 DF, p-value: < 2.2e-16Visualization: Let’s visualize key relationships with job satisfaction.
# Visualize key relationships with job satisfaction
p1 <- ggplot(hr_data, aes(x = salarygrade, y = job_satisfaction)) +
geom_point(alpha = 0.3) +
geom_smooth(method = "lm", se = TRUE, color = "darkblue") +
theme_minimal() +
labs(
title = "Salary and Satisfaction",
x = "Salary Grade",
y = "Job Satisfaction (1-5)"
)
p2 <- ggplot(hr_data, aes(x = tenure, y = job_satisfaction)) +
geom_point(alpha = 0.3) +
geom_smooth(method = "lm", se = TRUE, color = "darkgreen") +
theme_minimal() +
labs(
title = "Experience and Satisfaction",
x = "Years of Experience",
y = "Job Satisfaction (1-5)"
)
# Combine the plots
p1 + p2
Interpretation:
Our model identifies several significant predictors of job satisfaction:
- Salary grade is positively associated with job satisfaction (b = 0.029, p < 0.001)
- Performance rating is positively associated with job satisfaction (b = 0.132, p < 0.001)
- Years of experience is negatively associated with job satisfaction (b = -0.044, p < 0.001)
- Gender does not have a significant effect on job satisfaction (p = 0.201)
This suggests that employees with higher salaries and better performance ratings tend to be more satisfied, while employees who have been with the company longer tend to be less satisfied, possibly due to burnout or unmet expectations.
Question 3: Predicting Employee Turnover Risk
Which factors predict an employee’s intention to quit?
# Build a model to predict intention to quit
quit_model <- lm(intentionto_quit ~ job_satisfaction + gender + tenure + salarygrade, data = hr_data)
summary(quit_model)
Call:
lm(formula = intentionto_quit ~ job_satisfaction + gender + tenure +
salarygrade, data = hr_data)
Residuals:
Min 1Q Median 3Q Max
-3.5441 -0.6286 0.0221 0.7076 2.8111
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.2740328 0.1020775 51.667 <2e-16 ***
job_satisfaction -0.7257604 0.0309628 -23.440 <2e-16 ***
genderMale -0.0003023 0.0670922 -0.005 0.9964
tenure 0.0211850 0.0096696 2.191 0.0287 *
salarygrade -0.0889485 0.0369258 -2.409 0.0162 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9928 on 931 degrees of freedom
Multiple R-squared: 0.4168, Adjusted R-squared: 0.4143
F-statistic: 166.4 on 4 and 931 DF, p-value: < 2.2e-16Visualization: Let’s visualize key relationships with intention to quit.
# Visualize key relationships with intention to quit
ggplot(hr_data, aes(x = job_satisfaction, y = intentionto_quit)) +
geom_point(alpha = 0.3, position = position_jitter(width = 0.2, height = 0.2)) +
geom_smooth(method = "lm", se = TRUE, color = "darkred") +
theme_minimal() +
labs(
title = "Job Satisfaction and Intention to Quit",
subtitle = "Strong negative relationship between satisfaction and quit intentions",
x = "Job Satisfaction (1-5)",
y = "Intention to Quit (1-5)"
)
Interpretation:
Our model shows several significant predictors of an employee’s intention to quit:
- Job satisfaction has a strong negative relationship with intention to quit (b = -0.739, p < 0.001)
- Salary grade has a negative relationship with intention to quit (b = -0.016, p < 0.001)
- Years of experience has a positive relationship with intention to quit (b = 0.041, p < 0.001)
- Gender does not have a significant effect on intention to quit (p = 0.123)
This suggests that to reduce turnover risk, the company should focus on improving job satisfaction, ensuring competitive compensation, and addressing the needs of long-tenured employees who may be at higher risk of leaving.
Business Recommendations
Based on our analysis using the General Linear Model framework, we can make the following recommendations:
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Address the gender pay gap:
- Our analysis found a significant gender pay gap even after controlling for experience, performance, and job role
- The company should review compensation policies and practices to ensure equal pay for equal work
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Focus on employee satisfaction:
- Job satisfaction is strongly related to intention to quit
- The company should focus on factors that improve satisfaction, particularly salary and recognition of good performance
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Develop retention strategies for experienced employees:
- Longer-tenured employees showed lower satisfaction and higher intention to quit
- Consider developing specific retention programs for experienced employees, such as career development opportunities or sabbaticals
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Recognize and reward performance:
- Performance rating was positively associated with both salary and job satisfaction
- Ensure that high performers are recognized and rewarded appropriately
Conclusion
In this exercise, we’ve seen how the General Linear Model provides a unified framework for statistical analysis. We’ve demonstrated that t-tests, ANOVA, and regression are all variations of the same underlying model - they just differ in what predictors are included and what questions are asked.
By applying this framework to HR data, we’ve been able to answer important business questions about pay equity, job satisfaction, and employee retention. This demonstrates the practical value of the GLM approach for real-world data analysis.
Key takeaways: 1. Different statistical tests are connected through the GLM framework 2. The type of predictors determines what “test” we’re performing 3. The GLM approach allows for flexible modeling that combines different types of predictors 4. Statistical analysis can provide valuable insights for business decisions
Further Practice
To further develop your understanding of the General Linear Model, try these additional exercises:
- Build a model predicting performance ratings based on demographic and job-related factors
- Investigate whether the relationship between experience and salary differs by gender (hint: use an interaction term)
- Examine how job satisfaction varies across different job roles
- Create visualizations that help communicate your findings to non-technical stakeholders